This shows us that the exponential attenuation
in (32) is frequency dependent, and we can even plot the
real and imaginary parts of the wave number as functions of
frequency, as reported in Fig. 2.

Figure 2:
Imaginary and real part of the wave number as functions of frequency (
)

If the frequency range of interest is above a certain threshold, i.e.,
is small, we can obtain the following
relations from (47), by means of a Taylor expansion
truncated at the first term:

(47)

Namely, for sufficiently high frequencies, the attenuation can be considered to be constant and the dispersion relation can be considered to be the same as in a non-dissipative medium, as it can be seen from Fig. 2.

Still under the assumption of small losses, and truncating the Taylor
expansion of to the first term, we find that the wave
admittance (35) reduces to the two ``directional
admittances'':

(48)

where
is the admittance of the medium without losses, and
is a negative shunt reactance that accounts for losses.

The actual wave admittance of a one-dimensional medium, such as a tube, is while
is its paraconjugate in the analog domain. Moving to the discrete-time domain by means of a bilinear transformation, it is easy to verify that we get a couple of ``directional admittances'' that are related through (21).

In the case of the dissipative tube, as we expect, wave propagation is not lossless, since
. However, the medium is passive in the sense of section II-E, since the sum
is positive semidefinite along the imaginary axis.

The relations here reported hold for any one-dimensional resonator with frictional losses. Therefore, they hold for a certain class of dissipative strings and tubes. Remarkably similar wave admittances are also found for spherical waves propagating in conical tubes (see Appendix A).

The simulation of a length- section of lossy resonator can proceed according to two stages of approximation. If the losses are small (i.e.,
) the approximation (48) can be considered valid in all the frequency range of interest. In such case, we can lump all the losses of the section in a single coefficient
. The resonator can be simulated by the structure of Fig. 3, where we have assumed that the length is equal to an integer number of spatial samples.

At a further level of approximation, if the values of are even smaller we can consider the reactive component of the admittance to be zero, thus assuming
.

On the other hand, if losses are significant, we have to represent wave propagation in the two directions with a filter whose frequency response can be deduced from Fig. 2.
In practice, we have to insert a filter having magnitude and phase delay that are represented in Fig. 4 for different values of . From such filter we can subtract a contribution of linear phase, which can be implemented by means of a pure delay.

Figure 4:
Magnitude and phase delay introduced by frictional losses in a waveguide section of length , for different values of